korreleren

Korreleren, in statistics and signal processing, denotes a correlator — a device or algorithm that measures the similarity between two data sequences as a function of displacement or lag. It can be used to quantify how one signal relates to another or to identify repeated patterns within a signal.

For continuous signals x(t) and y(t), the cross-correlation is C_xy(τ) = ∫ x(t) y(t+τ) dt, and auto-correlation is C_xx(τ) = ∫ x(t) x(t+τ) dt. In discrete form, C_xy[n] = ∑ x[k] y[k+n]. Normalized correlation divides by the product of standard deviations to yield values in [-1, 1]. These definitions underpin how a korreleren assesses similarity across shifts in time or space.

Interpretation and properties: peaks in the correlation function indicate the best alignment between signals. The lag τ at which C_xy(τ) is maximal provides an estimate of the time delay between signals; the sharpness of the peak relates to how quickly similarity decays. In many cases, correlation functions satisfy symmetry properties such as C_xy(τ) = C_yx(-τ).

Applications: korrelere are used in synchronization and timing recovery, radar and sonar, wireless communications, and image or pattern matching. They support template matching, sequence detection, and analyses of coupling in neuroscience and physiology.

Implementation: hardware correlators rely on delay lines and parallel multipliers, while digital implementations use time-domain summation or FFT-based convolution to achieve efficient computation for long sequences. Normalization and windowing are common techniques to improve robustness in noisy data.

See also: correlation function, cross-correlation, auto-correlation, convolution, matched filter.